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Theorem qsqeqor 10573
Description: The squares of two rational numbers are equal iff one number equals the other or its negative. (Contributed by Jim Kingdon, 1-Nov-2024.)
Assertion
Ref Expression
qsqeqor ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵𝐴 = -𝐵)))

Proof of Theorem qsqeqor
StepHypRef Expression
1 qre 9571 . . . . . . 7 (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
21ad3antrrr 489 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 𝐴 ∈ ℝ)
3 simplr 525 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 0 ≤ 𝐴)
4 qre 9571 . . . . . . 7 (𝐵 ∈ ℚ → 𝐵 ∈ ℝ)
54ad3antlr 490 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 𝐵 ∈ ℝ)
6 simpr 109 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 0 ≤ 𝐵)
7 sq11 10535 . . . . . 6 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵))
82, 3, 5, 6, 7syl22anc 1234 . . . . 5 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵))
9 orc 707 . . . . 5 (𝐴 = 𝐵 → (𝐴 = 𝐵𝐴 = -𝐵))
108, 9syl6bi 162 . . . 4 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵𝐴 = -𝐵)))
11 oveq1 5857 . . . . . . 7 (𝐴 = 𝐵 → (𝐴↑2) = (𝐵↑2))
1211a1i 9 . . . . . 6 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 = 𝐵 → (𝐴↑2) = (𝐵↑2)))
13 oveq1 5857 . . . . . . . . 9 (𝐴 = -𝐵 → (𝐴↑2) = (-𝐵↑2))
1413adantl 275 . . . . . . . 8 (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 = -𝐵) → (𝐴↑2) = (-𝐵↑2))
15 qcn 9580 . . . . . . . . . 10 (𝐵 ∈ ℚ → 𝐵 ∈ ℂ)
16 sqneg 10522 . . . . . . . . . 10 (𝐵 ∈ ℂ → (-𝐵↑2) = (𝐵↑2))
1715, 16syl 14 . . . . . . . . 9 (𝐵 ∈ ℚ → (-𝐵↑2) = (𝐵↑2))
1817ad2antlr 486 . . . . . . . 8 (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 = -𝐵) → (-𝐵↑2) = (𝐵↑2))
1914, 18eqtrd 2203 . . . . . . 7 (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2))
2019ex 114 . . . . . 6 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 = -𝐵 → (𝐴↑2) = (𝐵↑2)))
2112, 20jaod 712 . . . . 5 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴 = 𝐵𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
2221ad2antrr 485 . . . 4 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴 = 𝐵𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
2310, 22impbid 128 . . 3 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵𝐴 = -𝐵)))
2417eqeq2d 2182 . . . . . 6 (𝐵 ∈ ℚ → ((𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
2524ad3antlr 490 . . . . 5 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
261ad3antrrr 489 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 𝐴 ∈ ℝ)
27 simplr 525 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 0 ≤ 𝐴)
28 qnegcl 9582 . . . . . . . . 9 (𝐵 ∈ ℚ → -𝐵 ∈ ℚ)
29 qre 9571 . . . . . . . . 9 (-𝐵 ∈ ℚ → -𝐵 ∈ ℝ)
3028, 29syl 14 . . . . . . . 8 (𝐵 ∈ ℚ → -𝐵 ∈ ℝ)
3130ad3antlr 490 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → -𝐵 ∈ ℝ)
32 simpr 109 . . . . . . . 8 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 𝐵 ≤ 0)
334le0neg1d 8423 . . . . . . . . 9 (𝐵 ∈ ℚ → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
3433ad3antlr 490 . . . . . . . 8 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
3532, 34mpbid 146 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 0 ≤ -𝐵)
36 sq11 10535 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (-𝐵 ∈ ℝ ∧ 0 ≤ -𝐵)) → ((𝐴↑2) = (-𝐵↑2) ↔ 𝐴 = -𝐵))
3726, 27, 31, 35, 36syl22anc 1234 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (-𝐵↑2) ↔ 𝐴 = -𝐵))
38 olc 706 . . . . . 6 (𝐴 = -𝐵 → (𝐴 = 𝐵𝐴 = -𝐵))
3937, 38syl6bi 162 . . . . 5 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (-𝐵↑2) → (𝐴 = 𝐵𝐴 = -𝐵)))
4025, 39sylbird 169 . . . 4 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵𝐴 = -𝐵)))
4121ad2antrr 485 . . . 4 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴 = 𝐵𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
4240, 41impbid 128 . . 3 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵𝐴 = -𝐵)))
43 0z 9210 . . . . . 6 0 ∈ ℤ
44 zq 9572 . . . . . 6 (0 ∈ ℤ → 0 ∈ ℚ)
4543, 44ax-mp 5 . . . . 5 0 ∈ ℚ
46 qletric 10187 . . . . 5 ((0 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (0 ≤ 𝐵𝐵 ≤ 0))
4745, 46mpan 422 . . . 4 (𝐵 ∈ ℚ → (0 ≤ 𝐵𝐵 ≤ 0))
4847ad2antlr 486 . . 3 (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) → (0 ≤ 𝐵𝐵 ≤ 0))
4923, 42, 48mpjaodan 793 . 2 (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵𝐴 = -𝐵)))
50 qnegcl 9582 . . . . . . . . . 10 (𝐴 ∈ ℚ → -𝐴 ∈ ℚ)
51 qre 9571 . . . . . . . . . 10 (-𝐴 ∈ ℚ → -𝐴 ∈ ℝ)
5250, 51syl 14 . . . . . . . . 9 (𝐴 ∈ ℚ → -𝐴 ∈ ℝ)
5352ad3antrrr 489 . . . . . . . 8 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → -𝐴 ∈ ℝ)
54 simplr 525 . . . . . . . . 9 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 𝐴 ≤ 0)
551le0neg1d 8423 . . . . . . . . . 10 (𝐴 ∈ ℚ → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
5655ad3antrrr 489 . . . . . . . . 9 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
5754, 56mpbid 146 . . . . . . . 8 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 0 ≤ -𝐴)
584ad3antlr 490 . . . . . . . 8 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 𝐵 ∈ ℝ)
59 simpr 109 . . . . . . . 8 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 0 ≤ 𝐵)
60 sq11 10535 . . . . . . . 8 (((-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((-𝐴↑2) = (𝐵↑2) ↔ -𝐴 = 𝐵))
6153, 57, 58, 59, 60syl22anc 1234 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((-𝐴↑2) = (𝐵↑2) ↔ -𝐴 = 𝐵))
6261biimpd 143 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((-𝐴↑2) = (𝐵↑2) → -𝐴 = 𝐵))
63 qcn 9580 . . . . . . . . . 10 (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
64 sqneg 10522 . . . . . . . . . 10 (𝐴 ∈ ℂ → (-𝐴↑2) = (𝐴↑2))
6563, 64syl 14 . . . . . . . . 9 (𝐴 ∈ ℚ → (-𝐴↑2) = (𝐴↑2))
6665adantr 274 . . . . . . . 8 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (-𝐴↑2) = (𝐴↑2))
6766eqeq1d 2179 . . . . . . 7 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((-𝐴↑2) = (𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
6867ad2antrr 485 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((-𝐴↑2) = (𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
69 negcon1 8158 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
7063, 15, 69syl2an 287 . . . . . . . 8 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
71 eqcom 2172 . . . . . . . 8 (-𝐵 = 𝐴𝐴 = -𝐵)
7270, 71bitrdi 195 . . . . . . 7 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (-𝐴 = 𝐵𝐴 = -𝐵))
7372ad2antrr 485 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → (-𝐴 = 𝐵𝐴 = -𝐵))
7462, 68, 733imtr3d 201 . . . . 5 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) → 𝐴 = -𝐵))
7574, 38syl6 33 . . . 4 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵𝐴 = -𝐵)))
7621ad2antrr 485 . . . 4 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴 = 𝐵𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
7775, 76impbid 128 . . 3 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵𝐴 = -𝐵)))
7852ad3antrrr 489 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → -𝐴 ∈ ℝ)
79 simplr 525 . . . . . . . 8 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐴 ≤ 0)
8055ad3antrrr 489 . . . . . . . 8 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
8179, 80mpbid 146 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 0 ≤ -𝐴)
8230ad3antlr 490 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → -𝐵 ∈ ℝ)
83 simpr 109 . . . . . . . 8 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐵 ≤ 0)
8433ad3antlr 490 . . . . . . . 8 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
8583, 84mpbid 146 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 0 ≤ -𝐵)
86 sq11 10535 . . . . . . 7 (((-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴) ∧ (-𝐵 ∈ ℝ ∧ 0 ≤ -𝐵)) → ((-𝐴↑2) = (-𝐵↑2) ↔ -𝐴 = -𝐵))
8778, 81, 82, 85, 86syl22anc 1234 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((-𝐴↑2) = (-𝐵↑2) ↔ -𝐴 = -𝐵))
8865, 17eqeqan12d 2186 . . . . . . 7 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((-𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
8988ad2antrr 485 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((-𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
9063ad3antrrr 489 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐴 ∈ ℂ)
9115ad3antlr 490 . . . . . . 7 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐵 ∈ ℂ)
9290, 91neg11ad 8213 . . . . . 6 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → (-𝐴 = -𝐵𝐴 = 𝐵))
9387, 89, 923bitr3d 217 . . . . 5 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵))
9493, 9syl6bi 162 . . . 4 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵𝐴 = -𝐵)))
9521ad2antrr 485 . . . 4 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴 = 𝐵𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
9694, 95impbid 128 . . 3 ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵𝐴 = -𝐵)))
9747ad2antlr 486 . . 3 (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) → (0 ≤ 𝐵𝐵 ≤ 0))
9877, 96, 97mpjaodan 793 . 2 (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵𝐴 = -𝐵)))
99 qletric 10187 . . . 4 ((0 ∈ ℚ ∧ 𝐴 ∈ ℚ) → (0 ≤ 𝐴𝐴 ≤ 0))
10045, 99mpan 422 . . 3 (𝐴 ∈ ℚ → (0 ≤ 𝐴𝐴 ≤ 0))
101100adantr 274 . 2 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (0 ≤ 𝐴𝐴 ≤ 0))
10249, 98, 101mpjaodan 793 1 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵𝐴 = -𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 703   = wceq 1348  wcel 2141   class class class wbr 3987  (class class class)co 5850  cc 7759  cr 7760  0cc0 7761  cle 7942  -cneg 8078  2c2 8916  cz 9199  cq 9565  cexp 10462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-2 8924  df-n0 9123  df-z 9200  df-uz 9475  df-q 9566  df-rp 9598  df-seqfrec 10389  df-exp 10463
This theorem is referenced by:  4sqlem10  12326
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